4.3.1 Basic Kelly Ⓜodel

From the papers Niko Marttinen (2001)1919 Kindly refer to 1th paper in Reference for industry knowdelege and academic research portion for the paper. and Jeffrey Alan Logan Snyder (2013)2020 Kindly refer to 2nd paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References both applying Full-Kelly,Half-Kelly and also Quarter-Kelly models which similar with my previous Kelly-Criterion model ®γσ, Eng Lian Hu 20142121 Kindly refer to 4th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References which had applied it and generates an impressive return. but enhanced. Niko Marttinen (2001) has concludes that the basic Kelly criterion generates the highest returns in long run compare to fractional Kelly models.

paper 4.3.1.1 : Niko Marttinen (2001)

diagram 4.3.0 get the idea from above paper which is count the odds price offers by bookmakers into calculation. My previous Odds Modelling and Testing Inefficiency of Sports Bookmakers odds modelling will be conducted further enhanced beyond next few years. In this staking model, I will also use the idea to measure the weakness of bookmakers but also enhanced our staking strategics. Meanwhile, Application of Kelly Criterion model in Sportsbook Investment2222 We can know from Part I where we can easily make profit from bookmakers but the Part II will enhanced to increase the profit and also the money management. will use a basket of bookmakers’ odds price to simulate it.

video 4.3.1.1 : Using Kelly Criterion for Trade Sizing

video 4.3.1.2 : Option Trading - The Kelly criterion formula: Mazimize your growth rate & account utility

video 4.3.1.3 : The Kelly criterion for trading options

To achieve the level of profitable betting, one must develop a correct money management procedure. The aim for a punter is to maximize the winnings and minimize the losses. If the punter is capable of predicting accurate probabilities for each match, the Edward O. Thorp (2006)2323 Kindly refer to 6th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References has proven to work effectively in betting. It was named after an American economist John Kelly (1956)2424 Kindly refer to 26th paper in Reference for industry knowdelege and academic research portion for the paper. in 7.4 References and originally designed for information transmission. The Kelly criterion is described below:

figure 4.3.1.1 : Kelly criterion formula.

\[S = \frac{\rho_{EM} \times BK_{Decimal\ odds} - 1} {BK_{HK\ odds}} \cdots equation\ 4.3.1.1\]

• Where \(S\) is the stake expressed as a fraction of one’s total bankroll.
• \(\rho_{EM}\) is probability of an event to take place and
• while \(BK_{Decimal\ odds}\) is decimal odds (decimal odds the return rates with capital stakes) and \(BK_{HK\ Odds}\) (HK odds is the net profit rates without capital stakes) for an event offered by the bookmaker.

Due to HK odds or decimal odds start from range \((0,\infty]\) and return will be \([0,\infty]\), therefore logarithmic function required. For Malay odds \([-1,1]\) no need logarithm. Here I switch from equation 4.3.1.1 to equation 4.3.1.2 as below.

\[log(S) = log(\rho_{EM}) + log(BK_{Decimal\ odds} - 1) - log(BK_{HK\ odds}) \cdots equation\ 4.3.1.2\]

Three important properties, mentioned by Hausch and Ziemba (1994)2525 You can refer to Efficiency of Racetrack Betting Markets (2008 Preface Edition) which is 29th paper in Reference for industry knowdelege and academic research portion for the paper. or Chapter 18 Efficiency of Sports and Lottery Betting Markets in FINANCE for further study about Hausch and Ziemba’s researchs. arise when using this criterion to determine a proper stake for each bet:

• It maximizes the asymptotic growth rate of capital
• Asymptotically, it minimizes the expected time to reach a specified goal
• It outperforms in the long run any other essentially different strategy almost surely

figure 4.3.1.2 : Example of application Kelly criterion.

The criterion is known to economists and financial theorists by names such as the geometric mean maximizing portfolio strategy, the growth-optimal strategy, the capital growth criterion, etc. We will now show that Kelly betting will maximize the expected log utility for sports-book betting.

The rRates value from table above excludes the Cancelled bets. By refer to equation 4.3.1.2, now we fit the adge value from equation 4.1.1 into it to get the rEMProbB2 and rEMProbL2 with known staked value \(S\)2727 Although the result will not be accurate due to the we mention at first, the firm A will not only place bets via only agent A. Let say edge of 0.10 and 0.20 also placed maximum bet HKD40000 but the firm A might placed different amount through other agency based on different edge. However based on the stakes we can reverse the optimal EM Odds. to replace the existing EM value. 2828 Initially think of linear modelling and get the mean value, the positive standard deviation value will be counted as edge range and the residuals value will be the different within the stakes across the leagues. It will similar with proportional staking model as states in paper Good and bad properties of the Kelly criterion by MacLean, Thorp and Ziemba (2010) and concludes that the Full-Kelly model is the best model for long run, you can refer to the reference in Kelly Criterion - Part II for further understanding.

\[log(\rho_{EM}) = log(S) + log(BK_{HK\ odds} + 1) - log(BK_{Decimal\ odds}) \cdots equation\ 4.3.1.3\]

Although the Kelly model is very simple, but we need to seperates the staking based on different leagues or time range to make it applicable to real world. I don’t pretend to know the correct model again but guess the applicable model by testing few models and choose the best among them.

We try to apply the equation 4.3.1.3 to get the Kelly stakes for every single soccer match.

Due to there have few reference papers conducting few staking strategics and concludes full Kelly model is the best fit and most profitable along the long term investment, here I try to simuulate the half-Kelly and also quadruple-Kelly, as well as double-Kelly staking model etc and get the optimal weighted control parameter.2929 There has a reference paper in section 2 of Application of Kelly Criterion model in Sportsbook Investment has compare few models as well and also provides the pro-and-con of Kelly model in investment. However, Kelly model will be the best across the long term investment. Besides, there have few papers doing research and also critic on the Kelly model in investment in financial market and also betting market (includes the rebates of the credit market as well), PIMCO’s fund manager Bill Gross who manage more than one trillion USD funds applied Kelly model for portfolio, George Soros and Warren Buffet also applied similar theoty or method with Kelly although there has no evidence to proof it. You are feel free to know in later section 4.5 Staking Ⓜodel and Ⓜoney Ⓜanagement. For further details kindly refer to Application of Kelly Criterion model in Sportsbook Investment.

Fractional Kelly models are the weight function for Kelly criterion When we talk about weight function in Kelly model. A Response to Professor Paul A Samuelson’s Objections to Kelly Capital Growth Investing has talk about the investment portfolio and compare the double-Kelly, full-Kelly, half-Kelly, quadruple-Kelly and also proportional betting across different stages of iterations and concludes that the full-Kelly will be the best fit and growth beyond the ages. Well, fractional-Kelly (means double-Kelly, half-Kelly and quadruple-Kelly but not full-Kelly model) models will be elastics and lesser will be more conservative and double-Kelly will be very risky and eventually going to bankcrupt due to the staking leverages ratio is twice of full-Kelly and over the sustainability of capital. and For further details kindly refer to Application of Kelly Criterion model in Sportsbook Investment. Therefore in last basic Kelly we use the full-Kelly within same leagues but due to there has different levels of risk setting across different soccer leagues. Therefore a weight function needed to make the staking strategy flexible, and it is term as Kelly portfolio to diversified the investment.

4.3.2 Fractional Kelly Ⓜodel

Now we try to fit a weight function into basic Kelly model to be fractional Kelly model. I try to use log to test the maximum value of weight parameter. You can just simply use \(w = \frac{1}{4}\) or \(log(w) = \frac{1}{2}\) while \(w\) is a vector. Please be mind that the value greater than 1 will be risky since involve leverage and lesser will be more conservative.

\[log(w_{i}) + log(\rho_{i}) \cdots equation\ 4.3.2.1\]

From Niko Marttinen (2001), we can know the full-Kelly generates couple times profit compare to fractional Kelly-models. However there has two points need to be enhanced.

• The high risk at the beginning period of investment.
• Test on different level of edge and concludes that the 145% generates the highest return.

Below Fabián Enrique Moya (2012) also test the fractional Kelly models with diversify money management methods.

paper 4.3.2.1 : Fabián Enrique Moya (2012)

table 4.3.2.1 : 177 x 6 : League stakes profiling of firm A year 2011~2015.

Above league risk profile suppose to stores the maximum bet for every single league but I only randomly select 6 leagues as sample. However due to I’ve not yet write a function for real time API3030 There are a lot of real time XML odds price and staking softwares similar with 4lowin2 which was states at the begining section in Part I with operators and test the maximum stakes per bet therefore here I reverse the mean value as the baseline stakes for every single league with a certain range of standard deviation for resampling simulation in later section.

4.3.3 Weighted Fractional Kelly Ⓜodels

In previous section I measure the data from 2011~2015 as static analysis. Well, now I try to seperates as annum data base and get the optimal weight value for next year use. Due to I dont know if the weight function is needed for staking models since sports consultancy firm had applied Poison models weith weight function. As we know a dice will have \(\frac{1}{6}\) chance to open one of the outcome, however theoretical probabilities doesn’t not correct since there have papers which applied bernoulli distribution to test the outcome with a certain iteration, the outcome might be 0.499 and 0.501 for over under but not 0.5 for each, that is due to the some effect like the balance of the dice, the flat level of the table, the wind, momentum and etc. I don’t pretand to know and only simulate it by obseravation to guess the optimal value.

Due to fractional Kelly model is independent models (for example : half-Kelly will be half-Kelly staking model, and full-Kelly will be only full-Kelly model across the years as we made comparison in section [4.3.2 Fractional Kelly odel].), now we need to make it weighted fractional model. Similar with my prevous Rmodel which applied on Poisson model. Due to the calculation of the settlement and result on the win and loss of Asian Handicap is different with Fixed odds, the probabilities of the outcome will be descrete and the measurement of the likelihood result required in order to maximize the profit. here we need to add an additional parameter controller to adjust the staking amount on every single match.

Now I try to will simulate an enhanced Kelly model on staking which take the effect of the outcome of the result into calculation below controller parameter \(\phi(r)\) fit into \(equation\ 4.3.3.1\) to control the leverage ratio.3131 similar theory apply on investment portfolio while it might turn to be nested controller parameters across different soccer leagues.

\[\phi(r) = exp(w_{i}\rho_{i}) \cdots equation\ 4.3.3.1\] Where \(X = x_{i,2,3...n}\) is the original staking amount by Kelly model. Meanwhile, the \(r\) value is the optimal parameter controller for staking.

\[r\begin{Bmatrix} =& Win\\ =& Half\ Win\\ =& Push\\ =& Half\ Loss\\ =& Loss\end{Bmatrix} \cdots equation\ 4.3.3.2\]

Here I try to diversified the weight parameters on equation 4.3.3.2. The first year data will be the baseline for further years analysis. The \(\phi(r)\) function is a contant variable which using previous years’s data to apply on current year staking model. You can also use other method to find yours.

Due to we unable foreseen the result before a soccer match started, here I tried to categorise from -1, 0.75, -0.5… 1 as a set of handicap.

table 4.3.3.1 : 5 x 8 : The static weight parameter from year 2011~2015.

Above table 4.3.3.13232 table 4.3.3.1 is wrong due to we cannot foreseen the result of a soccer match before kick-off, here I rewrote two kind of weight functions for handicap parameters just a static weighted parameter or constant across a year, you can simulate by apply the Epectation Maximization to get a dynamic vector of weight parameters across the soccer matches. The later section will conduct a monte carlo simulation from 2011 until 2015 to get the best fit outcome.

4.3.4 Dynamic Fractional Kelly Ⓜodel

4.3.5 Bank Roll

There has few points we need to consider, there the we need to retrieve the initial investment capital \(BR\):

• risk averse from ruin3333 The daily lost cannot over the investment fund, otherwise will be bankrupt before growth, Kelly model is sort of risky at the beginning period but bacame stable as time goes by.
• Initial invested capital
• The differences of time zone (British based sports consultancy firm A)
• The financial settlement time of Asian operators (daily financial settlement time 12:00PM Hong Kongnese GMT+8 Credit market with rebates)3434 Here I follow the kick-off time from GMT+8 1200 until next morning 1159 (or the American Timezone which is GMT - 4) considered as a soccer betting date. Re-categorise the soccer financial settlement date. Due to I have no the history matches dataset from bookmakers. The scrapped spbo time is not stable (always change, moreover there just an information website) where firm A is the firm who placed bets with millions HKD (although the kick-off time might also changed after placed that particular bet), therefore I follow the kick-off time of the firm A.

graph 4.3.5.1 : Sample data of bank roll and fund growth for basic Kelly model. ($1 = $10,000)

Above Graph is a basic Kelly model, we can know the initial fund size is not united.

Due to our bank roll cannot be less than 0, otherwie will be ruined. Therefore I added the initial balance of the account from the min value of variable SPL which is the balance before place bets must be more than 0. Otherwise unable to place bets. 4.5.1 Risk Management will united the initial fund size and Kelly portion across the league profiles.

The file BankRoll.csv states the profit and loss of the staking. You can see the

As I mentioned at the begining of the research paper, the stakes only reflects the profit and loss of agency A but not firm A. Firm A might have deal with 10~50 or even more agencies and the data from year 2011 is not the initial investment year. You are feel free to download the file. We will discuss the inventory management to reduce the risk.

graph 4.3.5.1 has Event label to marking a specific event on a specifi date or time while I just leave it and only mark on high volatily event dates. From BankRoll.csv we observe the end of soccer sesson in May 2011 dat %>% filter(DateUS >= '2011-05-14' & DateUS <= '2011-05-21') has a seriourly crash. We can investigate more details about the loss matches from the data (or filter the range of the bets in the data table inside Part I).

4.4.1 Niko Marttinen (2001)

Data has been collected over the last four seasons in the English Premier League. These include 1997-1998, 1998-1999, 1999-2000 and 2000-2001 seasons. We have also collected the season 2000-2001 data from the main European football betting leagues, such as English Division 1, Division 2 Division 3, Italian Serie A, German Bundesliga and Spanish Primera Liga…

quote 4.4.1.1 : the dataset for the studies (source : Niko Marttinen (2001)).

Niko Marttinen (2001)3535 Kindly refer to 1th paper in Reference for industry knowdelege and academic research portion for the paper. has enhanced the Dixon and Coles (1996) which are :

• Basic Poisson model : Independence Poisson model for both home and way teams with a constant home advantage parameter.
• Independent home advantages model : Seperate the home advantage parameter depends on the teams accordingly.
• Split season model : Split a soccer league season to be 1st half and 2nd half season.
• 5. Scores plus Poisson model.

From above models, the author has compare the efficiency and the best fit model for scores prediction as below.

figure 4.4.1.1 : Comparison of various Poison models (source : Niko Marttinen (2001)).

From figure 4.4.1.1 above, the author compare the deviance of the models3636 Kindly refer to Generalized Linear Models in R, Part 2: Understanding Model Fit in Logistic Regression Output, devianceTest and Use of Deviance Statistics for Comparing Models to learn about the method of comparison.

figure 4.4.1.2 : Comparison of various mixed Poison models II (source : Niko Marttinen (2001)).

figure 4.4.1.3 : Comparison of various mixed Poison models III (source : Niko Marttinen (2001)).

From above models, the author list the models and states that even though pick the worst model among the models still more accurate than bookmaker while E(Score)&Dep&Weighted is the best.

figure 4.4.1.4 : Comparison of various odds modelling models (source : Niko Marttinen (2001)).

Besides, Niko Marttinen (2001) not only choose Poison model throughly as the odds modelling model but also compare to below models :-

• ELO ratings.
• multinomial ordered probit model.

He concludes that the multinomial ordered probit model is the best fit model but the software for fitting is not generally available. Meanwhile, the Poisson model is more versatile than probit logit model based on the dataset accross the European soccer leagues.3737 There has a lot of papers with regard to application of logit probit models on soccer betting, might read through and made comparison with my ®Model ®γσ, Eng Lian Hu (2016). I used to read though the logit probit and there has a complicated parameters setting for various effects like : wheather, players’ condition, couch, pitch condition and even though the distance travel and the players’ stamina modelling.

4.4.2 Dixon and Coles (1996)

Here we introduce the Dixon and Coles (1996) model and its codes. You are freely learning from below links if interest.

• Dixon and Cole’s Poisson regression R Packages
• Dixon and Coles Poisson model
• Dixon Coles model - Python
• Predicting Football Using R

Due to the soccer matches randomly getting from different leagues, and also not Bernoulli win-lose result but half win-lose etc as we see from above. Besides, there were mixed Pre-Games and also In-Play soccer matches and I try to filter-up the sample data to be only English soccer leagues as shinyApps. I don’t pretend to know the correct answer or the model from firm A. However I take a sample presentation Robert Johnson (2011)3838 Kindly refer to 23th paper in 7.4 References from one of consultancy firm which is Dixon-Coles model and omitted the scoring process section.

4.4.3 ®γσ, Eng Lian Hu (2016)

Below is my previous research paper which was more sophiscated than Dixon-Coles model. You can refer it and I will just omit the section as mentioned at the beginning section of this staking validation research paper.

paper 4.4.3.1 : ®γσ, Eng Lian Hu (2016)

By refer to 4.4.1 Niko Marttinen (2001), I’ve add the enhancement on Odds Modelling and Testing Inefficiency of Sports Bookmakers as future improvement while the weighted function might be enhanced soonly.

Here I cannot reverse computing from barely \(\rho_i^{EM}\) without know the \(\lambda_{ij}\) and \(\gamma\) values. Meanwhile, the staked matches is a descrete random soccer teams accross all leagues and tournaments. Therefore I just simply use reverse EM probabilities by mean value of edge in previous section Kelly.

\[X_{ij} = pois(\gamma \alpha_{ij} \beta_{ij} ); Y_{ij} = pois(\alpha_{ij} \beta_{ij}) \cdots equation\ 4.4.1\]

4.4.4 Combination Handicap

In order to minimzie the risk, I tried to validate the odds price range invested by firm A.3939 As I used to work in AS3388 which always take bets from Starlizard where they only placed bets within the odds price range from 0.70 ~ -0.70. They are not placed bets on all odds price in same edge. The sportbook consulatancy firms might probably not place same amount of stakes on same edge, lets take example as below :-

• \(Odds_{em}\) = 0.40 while \(Odds_{BK}\) = 0.50, The edge to firm will be 0.5 ÷ 0.4 = 1.25
• \(Odds_{em}\) = 0.64 while \(Odds_{BK}\) = 0.80, The edge to firm will be 0.8 ÷ 0.64 = 1.25

We know above edge is same but due to the probability of occurance an event/goal at 0.4 is smaller than 0.64. In 4.3.3 Weighted Fractional Kelly Ⓜodels I tried to use a weight function to measure the effect of Win-All, Win-Half, Push, Loss-Half, Loss and also Cancelled and there has generated a higher profit.

Again, I don’t pretend to know the correct models but here I try to simulate the occurance of the combination and independent handicaps by testing below distribution.

• categorise handicap set (from start of previous session until latest soccer matches as Weight function which has talked in 4.3.3 Weighted Fractional Kelly Ⓜodels)
• normal distribution
• binomial distribution
• Poison distribution

Due to there have 110 different weighted Kelly models I test, here I put the application of distribution in specific English soccer league which will compare with my Rmodel in Application of Kelly Criterion model in Sportsbook Investment - Part II.

Comparison Chart

figure 4.4.4.1 : Binomial vs. Poisson

figure 4.4.4.2 : Difference Between Binomial and Poisson Distribution

You can know about the use of the distribution through below articles.

• figure 4.4.4.1 : Binomial vs. Poisson
• figure 4.4.4.2 : Difference Between Binomial and Poisson Distribution
• Difference between Poisson and Binomial distributions
• Statistical Distributions (e.g. Normal, Poisson, Binomial) and their uses

figure 4.4.4.3 : Diagram of network among probability distributions

From above diagram we can know the network and relationship among probability distributions. Besides, we can try to refer to below probability distribution listing attach with R codes and examples for pratice :

• Probability Distributions in R (Stat 5101, Geyer)

figure 4.4.4.4 : R functions in probability distributions

Here I try to bootstrap/resampling the scores of matches of the dataset to test the Kelly model and get the mean/likelihood value. Boostrapping the scores and staking model will be falling in the following sections [4.5 Staking Ⓜodel and Ⓜoney Management] and 4.6 Expectation Ⓜaximization and Staking Simulation.

4.5.1 Risk Management

graph 4.5.1.1 : Sample data of candle stick chart for fund growth of firm A via agent A. ($1 = $10,000)

Above table shows the return rates of investment fund from firm A via agent A. We know the initial unvestment fund from $47,788,740.00 growth to $405,511,999.00 within year 2011-01-07, 2015-07-19 with a high return rates 848.55%. Well, I try to apply Kelly model for stakes management which is in previous section 4.3 Kelly Ⓜodel.

Galema, Plantinga and Scholtens (2008)4040 You are feel free to refer to Reference for industry knowdelege and academic research portion for the paper. in 7.4 References for further details introduce Fama-MacBeth regressions4141 FM-regression is a Capital asset pricing model (CAPM) which estimate the rates of return of an asset. for portfolio management. The method estimates the betas and risk premia for any risk factors that are expected to determine asset prices. The method works with multiple assets across time (panel data). The parameters are estimated in two steps:

• First regress each asset against the proposed risk factors to determine that asset’s beta for that risk factor.
• Then regress all asset returns for a fixed time period against the estimated betas to determine the risk premium for each factor.

Assumptions of CAPM

All investors:

• Aim to maximize economic utilities (Asset quantities are given and fixed).
• Are rational and risk-averse.
• Are broadly diversified across a range of investments.
• Are price takers, i.e., they cannot influence prices.
• Can lend and borrow unlimited amounts under the risk free rate of interest.
• Trade without transaction or taxation costs.
• Deal with securities that are all highly divisible into small parcels (All assets are perfectly divisible and liquid).
• Have homogeneous expectations.
• Assume all information is available at the same time to all investors.

Here I skip above section and will leave it as future study due to it will estimate below point which invilve in financial and accounting :

• active investors, inactive investors and potential investors.
• rates of return of an asset (which is my previous courses in Coursera Improving Business Finances and Operations but not yet complete the specialization)
• governmental taxes.

Martin Spann and Bernd Skiera (2009)4242 Kindly refer to 19th paper in Reference for industry knowdelege and academic research portion for the paper. under 7.4 References applied a basic probability sets on the draw games and also the portion of win and loss. The author simply measured the portion of the draw result with win/loss to get the edge to place a bet. However it made a loss on Italian operator Oddset due to the 25% high vigorish but profitable in 12%. Secondly, the bets placed on fixed odds but not Asian Handicap and also a fixed amount $100.

video 4.5.1.1 : Calculated Bets - computers, gambling, mathematical modeling to Win (part 4 of 4)

table 4.3.2.1 shows a risk portfolio for every single league which is roughly similar with Parimutuel Betting but a portion among the initial fund required. In order to equalise the intial fund size. next section I will united it and also application of resampling method to get the optimal league risk profile.

paper 4.5.1.1 : Magnus Erik Hvass Pedersen (2014)